THE SETS of MONOMORPHISMS and of ALMOST OPEN OPERATORS BETWEEN LOCALLY CONVEX SPACES 1. Introduction and Preliminaries Bounded B

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 12, Pages 3683–3690 S 0002-9939(01)06248-7 Article electronically published on June 27, 2001

THE SETS OF MONOMORPHISMS AND OF ALMOST OPEN OPERATORS BETWEEN LOCALLY CONVEX SPACES

JOSE´ BONET AND JOSE´ A. CONEJERO

(Communicated by Jonathan M. Borwein)

Abstract. If the set of monomorphisms between locally convex spaces is not empty, then it is an open subset of the space of all continuous and linear operators endowed with the topology of the uniform convergence on the bounded sets if and only if the domain space is normable. The corresponding characterization for the set of almost open operators is also obtained; it is related to the lifting of bounded sets and to the quasinormability of the domain space. Other properties and examples are analyzed.

1. Introduction and preliminaries Bounded below and almost open continuous linear operators between normed spaces, their relation with the topological divisors of zero in the normed algebra of all operators, and the approximate point spectrum have been extensively studied. We refer to the books of Berberian [3, Sections 56, 57], Harte [8], and the articles of Abramovich, Aliprantis and Polyrakis [2] and Harte [9]. In [2, Proposition 2.2] it is shown that the set of bounded below operators (or monomorphisms) between two normed spaces X and Y is open in the normed space of operators L(X, Y ). The corresponding result for almost open operators can be seen in [8, Theorem 3.4.3]. These results are extensions of the well-known fact (see e.g. [10, Theorem 18.12]) that the set of isomorphisms from a Banach space X onto a Banach space Y is an open subset of the Banach space L(X, Y ), a result which can be proved using the Neumann series for a linear operator. We refer to the recent article by Casazza and Kalton [5] for extensions of Paley-Wiener perturbation theory. In the case of continuous linear operators from a locally convex space E into itself, it was proved by Kasahara in 1972 in [12] that if the set of isomorphisms is open in the space of operators Lb(E) endowed with the topology of the uniform convergence on the bounded subsets of E, then the space E must be normable (see the quotation in M. Akkar [1]). If E is complete, in the terminology of topological algebras, Lb(E) is a Q-algebra if and only if E is normable.

Received by the editors May 2, 2000. 2000 Mathematics Subject Classiﬁcation. Primary 46A32, 46A03, 46H35, 47A05, 47L05. Key words and phrases. Bounded below operators, monomorphisms, almost open operators, locally convex spaces, quasinormable spaces. The authors were partially supported by the project DGESIC, PB97-0333. The second author was also supported by the Universidad Polit´ecnica de Valencia, grant 19980998.

c 2001 American Mathematical Society 3683

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The purpose of this note is to characterize completely when the set of monomorphisms and the set of almost open operators are open in the set of all operators L(E,F) between two locally convex spaces E and F for the topology of uniform convergence on the bounded subsets of E. The characterizations are obtained in Theorems 2.1 and 2.2. The role of the quasinormability of the space E is analyzed in Proposition 2.4 and Corollary 2.5. Theorem 2.6 studies the set of surjective and open operators between non-complete normed spaces. Several examples and propositions discuss other properties of these classes of operators. Our notation for Banach spaces, operator theory and locally convex spaces is standard and we refer to the books [3, 8, 10, 11, 15, 17, 18]. Unless explicitly mentioned, E and F denote Hausdorﬀ locally convex spaces. If E is a locally convex space, the set of continuous seminorms on E is denoted by cs(E). If p ∈ cs(E), the unit ball of this seminorm is denoted by Up := {x ∈ E | p(x) ≤ 1}. The family of all closed absolutely convex neighbourhoods of the origin in E is denoted by U0(E). The Minkowski functional of U ∈U0(E) is denoted by pU , and pU ∈ cs(E). If E is a normed space with the norm ||.||, its closed unit ball is denoted by UE := {x ∈ E |||x|| ≤ 1}. The set of all closed absolutely convex bounded subsets of a locally convex space E is denoted by B(E). If B ∈B(E), we denote by EB the normed space given by the linear span of B endowed with the norm deﬁned by the Minkowski functional pB of B.ThesetB ∈B(E)isa Banach disc if EB is a Banach space. The space E is called locally complete if every B ∈B(E) is a Banach disc. Every complete space is locally complete. As usual, the symbol Lb(E,F) denotes the locally convex space of all (always linear and continuous) operators from a locally convex space E into a locally convex space F , endowed with the locally convex topology of uniform convergence on the bounded subsets of E. A basis of neighbourhoods of zero of Lb(E,F)isgivenby the sets W(B,V ):={T ∈ L(E,F) | T (B) ⊆ V }

as B runs in B(E)andV runs in U0(F ). This topology is defined by the seminorms qB(T ):=sup{q(T (b)) | b ∈ B},forB ∈B(E)andq ∈ cs(F ). If E and F are normed, then Lb(E,F) is the usual normed space of operators. If E = F ,wewrite Lb(E). An operator T ∈ L(E,F) is called bounded below if for every p ∈ cs(E)there is q ∈ cs(F )withp(x) ≤ q(T (x)) for each x ∈ E. Compare with the definition in the case of normed spaces in [2, Definition 2.1], [3, p. 173] and [8, Definition 3.3.1]. Clearly every bounded below operator is injective. It is easy to see that an operator T ∈ L(E,F) is bounded below if and only if T is an isomorphism from E into F , i.e. T is a monomorphism in the sense of [11, vol. II, p. 2 ff.], i.e. T is linear continuous injective and open into its image. The set of all monomorphisms from E to F is denoted by mo(E,F). An operator T ∈ L(E,F) is called almost open,andwewriteT ∈ ao(E,F)if for every U ∈U0(E)thereisV ∈U0(F ) such that V ⊆ T (U), the closure taken in F . Clearly every almost open operator has dense range. Almost open operators are precisely the nearly open operators with dense range in the sense of Pták (see [11, vol. II, p. 24]). The case of almost open operators between normed spaces is studied in [8, p. 65 ff.]. The class of almost open operators between Fréchet spaces coincides with the class of surjective operators by the Banach-Schauder open mapping theorem (see [11, vol. I, p. 166] or [15, Chapter 8]). The relation between

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the properties of T ∈ L(E,F) mentioned above and its transpose T t : F 0 → E0 is analyzed in [3, 57.16–18], [8, 5.5.2–4], and [9, Section 2] for normed spaces E and F . The situation for locally convex spaces is more complicated; we refer to Dierolf and Zarnadze [6].

2. Results Theorem 2.1. Assume that mo(E,F) is not empty. The set mo(E,F) is open in Lb(E,F) if and only if E is normable.

Proof. First we suppose that mo(E,F) is open and non-empty. We select T0 ∈ mo(E,F), and a pair B ∈B(E)andV ∈U0(F ) such that every S ∈ L(E,F)with (S − T0)(B) ⊆ V satisfies that S ∈ mo(E,F). We claim that V ∩ T0(E) ⊆ T0(B), which implies that V ∩T0(E) is a bounded neighbourhood of zero in T0(E), whence E is normable by Kolmogorov’s theorem. We prove the claim by contradiction: Suppose that there is x ∈ E with T0(x) ∈ V \ T0(B), in particular x =6 0. We apply 0 the Hahn-Banach theorem to find v ∈ F such that v(T0(x)) = 1, |v(T0(b))| < 1for all b ∈ B. We define S : E → F by S(z):=T0(z) − v(T0(z))T0(x) for every z ∈ E. Clearly S ∈ L(E,F). For every b ∈ B we have (T0 −S)(b)=v(T0(b))T0(x) ∈ V .By assumption S ∈ mo(E,F). But S(x) = 0, and this implies that S is not injective, a contradiction. Now we assume that E is a normed space with a norm ||.||, and we fix T0 ∈ mo(E,F). By definition, there is q ∈ cs(F ) such that ||x|| ≤ q(T0(x)) for each ∈ ∈ { − ||| || ≤ }≤ 1 x E.IfS L(E,F) satisfies sup q(T0(x) S(x)) x 1 2 ,wehave 1 ||x|| ≤ q(T (x)) ≤ q(S(x)) + q(T (x) − S(x)) ≤ q(S(x)) + ||x|| for x ∈ E. 0 0 2 This implies ||x|| ≤ 2q(S(x)) for each x ∈ E and we conclude that S ∈ mo(E,F).

Theorem 2.2. Assume that ao(E,F) is not empty. The set ao(E,F) is open in Lb(E,F) if and only if the space F is normable and for every T ∈ ao(E,F) there is B ∈B(E) such that T (B) is a neighbourhood of the origin in F .

Proof. We suppose that ao(E,F)isopeninLb(E,F) and not empty. Let T be an arbitrary element of ao(E,F). By assumption there are B ∈B(E)andV ∈U0(F ) such that if S ∈ L(E,F)satisfies(S − T )(B) ⊆ V ,thenS ∈ ao(E,F). We claim that V ⊆ T (B), from where the necessity follows. Proceeding by contradiction, assume that there is y ∈ V \ T (B). By the Hahn-Banach theorem there is v ∈ F 0 with v(y)=1,|v(T (b))| < 1 for every b ∈ B. We define S ∈ L(E,F)byS(z):= T (z) − v(T (z))y.Ifb ∈ B,weget(T − S)(b)=v(T (b))y ∈ V . On the other hand, S(E) ⊆ ker v since v(S(z)) = v(T (z)) − v(T (z))v(y)=0,z∈ E. This implies that S(E) is not dense in F ,andS cannot be almost open. A contradiction. To prove the converse we fix T ∈ ao(E,F). By assumption F is normed and there is B ∈B(E) such that T (B) contains the unit ball UF of F , so that the operator Tˆ := T |EB : EB → F is almost open between these two normed spaces. By [8, ˆ ∈ || ˆ − ˆ|| ≤ Theorem 3.4.3], there is >0 such that if R L(EB,F)and R T L(EB,F ) , then Rˆ ∈ ao(EB ,F). Now W(B,UF ) is a neighbourhood of the origin in Lb(E,F).

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If S ∈ L(E,F)satisﬁes(S − T ) ∈W(B,UF ), then Sˆ := S|EB ∈ L(EB,F)satisﬁes || ˆ − ˆ|| ≤ ˆ ∈ ˆ S T L(EB ,F ) . Therefore S ao(EB ,F) and this yields that S(B)=S(B) is a neighbourhood in F .

The following consequence follows from the Banach-Schauder open mapping theorem and it extends [2, Theorem 3.1].

Corollary 2.3. Let E and F be Fr´echet spaces. Assume that there is a surjective operator from E onto F . The set of all surjective operators from E onto F is open in Lb(E,F) if and only if F is a Banach space and for every surjective operator T : E → F there is a bounded subset B of E such that T (B) is a neighbourhood in F .

Concerning the assumption in Theorems 2.1 and 2.2 and Corollary 2.3 it is worth mentioning that there are pairs of Banach spaces or Fréchet spaces (E,F) such that mo(E,F)=∅ and ao(E,F)=∅. This is well known for E = lp and F = lq with p =6 q (see [13, Chapter 2]). Vogt [19] and Bonet [4] investigated pairs of Fréchet spaces (E,F) such that every T ∈ L(E,F) maps a neighbourhood in E into a bounded subset of F . These pairs satisfy that mo(E,F)andao(E,F)areempty, if E and F are not normable. Not every surjective operator from a Fréchet space E onto a Banach space F satisfies that the unit ball of the Banach space F is contained inside the image of a bounded subset of E:ThereisaKöthe echelon space E = λ1(A)whichisaMontel space having a quotient isomorphic to F = l1; see [11, 31.5] or [15, 27.21 and 27.22]. The quotient map q : E → F does not lift bounded sets since the bounded subsets of E are relatively compact. The class of quasinormable spaces was introduced and studied by Grothendieck. It contains Banach and nuclear spaces, and it is also stable by quotients. Every DF-space in the sense of Grothendieck is quasinormable. A locally convex space is called quasinormable if for every U ∈U0(E)thereisV ∈U0(E) such that for every >0thereisB ∈B(E)withV ⊆ B + U. Quasinormable Fréchet spaces and their relevance for the lifting of bounded sets can be seen in [15, Chapter 26] and [17, Section 8.3]. Our next result extends Miñarro [14, Theorem 1]. We thank A. Peris for providing us with this elementary proof.

Proposition 2.4. (a) Let E be a quasinormable space and let F be a normed space. If T ∈ ao(E,F), then there is B ∈B(E) such that UF ⊆ T (B). (b) If q is a surjective open operator between a locally complete quasinormable space E and a Banach space F , then there is B ∈B(E) such that q(B) is a neighbourhood in F .

Proof. Part (b) is an easy consequence of (a) and the Banach-Schauder open mapping theorem, since EB is a Banach space. ∈U ⊆ 1 We prove (a): Since T is continuous we find U 0(E)withT (U) 4 UF .As E is quasinormable, we find V ∈U0(E) such that for all µ>0thereisB ∈B(E) with µV ⊆ (B + U). We apply that T is almost open to find λ>0withUF ⊂ ⊂ 1 ∈B ⊆ λT (V ) λT (V )+ 4 UF . We select C (E)withλV C + U to conclude 1 1 1 U ⊆ λT (V )+ U ⊆ T (C)+T (U)+ U ⊆ T (C)+ U . F 4 F 4 F 2 F

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Proceeding by recurrence we have, for each n ∈ N, − nX1 1 1 U ⊆ 2−kT (C)+ U ⊆ T (2C)+ U . F 2n F 2n F k=0

This yields UF ⊆ T (2C).

Corollary 2.5. (a) If E is a quasinormable space and F is a normed space, then ao(E,F) is an open subset of Lb(E,F). (b) If E is a locally complete quasinormable space and F is a Banach space, then the set of surjective and open operators from E onto F is open in Lb(E,F). Corollary 2.5 extends the results of [8, Theorem 3.4.3] and of [2, Theorem 3.1]. Simple examples in [2, p. 460] show that 2.5 b) does not hold for normed spaces. These ideas are exploited in our next result. Theorem 2.6. (a) Every infinite-dimensional separable Banach space X contains a dense subspace E such that the set of surjective and open operators onto E is not an open subset of the normed space L(E). (b) Every infinite-dimensional Banach space X contains a proper dense subspace F such that the set of surjective and open operators onto F is an open subset of the normed space L(F ). ∞ ⊆ Proof. a) By a result of Ovsepian and Pelczy´ nski [16], there are sequences (xn)n=1 || || { | ∈ N} ∞ ⊆ 0 || || ≤ X, xn =1,span xn n = X and (un)n=1 X with un 2 such that um(xn)=δnm (Kronecker’s delta). We set E := span(x1,x2,...). For >0we define T : E → E by ∞ X 1 T (x):=x − u (x)x . 2n+2 n n+1 n=1

The operator T is clearly well-deﬁned and continuous by the properties of the ∞ ∞ ∈ || || ≤ biorthogonal sequences (xn)n=1 and (un)n=1.Ifx E and x 1, then ∞ X 1 ||(I − T )(x)|| ≤ ||u || ||x || ≤ . 2 2n+1 n n+1 n=1 P k ∈ 6 For each x := i=1 αixi E with αk =0,wehave 1 u (T (x)) = − α =06 . k+1 2 2k+1 k

This implies that x1 ∈/ T(E). The proof is complete in this case. b) Let X be an inﬁnite-dimensional Banach space, and let u : X → K be a non-continuous linear form. Then F := ker u is a dense hyperplane of X.Weﬁrst show the following observation. Remark 2.7. Let g ∈ L(X) be a surjective operator such that g(F ) ⊆ F .Let g˜ := g|F ∈ L(F ) be the restriction of g to F . Then: (i) g(X \ F ) ⊆ X \ F . (ii)g ˜ is surjective onto F . (iii) ker g =ker˜g.

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Indeed, to show (i) suppose there is z ∈ X\F with g(z) ∈ F .SinceX = F ⊕span(z), we conclude g(X) ⊆ g(F ) ⊕ span(g(z)) ⊆ F =6 X, which is a contradiction, since g is surjective in X. To show (ii), we pick y ∈ F .Sinceg is surjective there is z ∈ X with g(z)=y ∈ F .By(i)z ∈ F andg ˜ is surjective. Finally to prove (iii) we first observe that kerg ˜ ⊆ ker g.Ifx ∈ ker g ⊆ X,wehaveg(x)=0∈ F .By(i)x ∈ F and g(x) = 0, i.e. x ∈ kerg ˜. We continue with the proof of part (b). We fix T ∈ L(F ) which is surjective and open (i.e. a surjective homomorphism in the sense of [11, Chapter 32]). By density there is a unique extension Tˆ ∈ L(X)withTˆ|F = T . We apply [11, 32.5(3)] to conclude that Tˆ is a homomorphism in the Banach space X.ConsequentlyTˆ(X) is a closed subspace of X.SinceF ⊆ Tˆ(X), we have that Tˆ is surjective. By [2, ˆ Theorem 3.1], there is >0 such that if R ∈ L(X)satisfies||R − T ||L(X) ≤ , then R is surjective. Let S ∈ L(F )satisfy||S − T ||L(F ) ≤ . The unique extension ˆ ˆ ˆ ˆ S ∈ L(X)ofS to X satisfies ||S − T ||L(X) ≤ , hence S is surjective onto X.We apply Remark 2.7 to g := Sˆ,˜g = g|F = S, to conclude that S ∈ L(F ) is surjective onto F ,andkerSˆ =kerS. By [17, 2.6.18] S is a homomorphism in F . Therefore S ∈ L(F ) is surjective and open. In [2, Theorems 3.3 and 3.4] it is proved that, for Banach spaces E and F , the set of isomorphisms from E onto F coincides with mo(E,F) ∩ ao(E,F)and with ao(E,F) ∩ mo(E,F), the closures taken in Lb(E,F). Our examples below show that these results do not hold for Fréchet spaces E,F.Wedenotebyω the countable product of copies of the scalar field. Example 2.8. The forward shift operator F : ω → ω defined as

F (x1,x2,x3,...)=(0,x1,x2,x3,...) is a monomorphism which is not surjective. The operator −F belongs to the closure − 1 − ∞ of ao(ω,ω). In fact Hn := F + n I converges to F as n tends to in Lb(ω,ω), − 1 ∞ ∈ and F + n I is a surjective operator: indeed, for y =(yj)j=1 ω, we deﬁne ∞ ∈ 2 j−1 j x =(xn)n=1 ω by xj := nyj + n yj−1 + ...+ n y2 + n y1. Clearly Hn(x)=y. Consequently mo(ω,ω) ∩ ao(ω,ω) is not contained in the set of isomorphisms onto ω. Example 2.9. The backward shift operator B : ω → ω deﬁned as

B(x1,x2,x3,...)=(x2,x3,...)

is surjective and not injective, but it belongs to the closure of mo(ω,ω)inLb(ω,ω). In fact, we deﬁne Hn : ω → ω, Hn(x1,x2,x3,...)=(x2,x3,... ,xn,x1,xn+1,...). Clearly Hn(x)convergestoB(x)asn tends to ∞ for each x ∈ ω. By the Banach- Steinhaus theorem [11, 39.5(1)], Hn tends to B as n tends to ∞ on every precompact subset of ω.Sinceω is a Montel space Hn converges to B as n goes to ∞ in Lb(ω,ω). Consequently ao(ω,ω) ∩ mo(ω,ω) is not contained in the set of isomorphisms onto ω. More can be said about a sequence of operators which converges in a very strong sense to a surjective isomorphism T ∈ L(E). We need the following extension of the classical result for Banach spaces which is due to Garnir, De Wilde and Schmets [7, p. 346]. Its proof uses the Neumann Series. An operator T ∈ L(E,F) is called

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bounded (resp. compact )ifthereisU ∈U0(E) such that T (U) is bounded (resp. relatively compact) in F . Theorem 2.10 (Garnir, De Wilde, Schmets). Let E be a complete locally convex space. Let T ∈ L(E) be an operator such that there is U ∈U0(E) with (I − T )(U) bounded in E and (I −PT )(U) ⊆ αU for some 0 <α<1.ThenT is an isomorphism −1 ∞ − k − onto E,andT = k=0(I T ) , the series converging in Lb(E).If(I T ) is compact no completeness assumption is needed on E. Proposition 2.11. Let E be a complete locally convex space, T ∈ L(E) an iso- ∞ ∈B ∈U morphism, and (Tn)n=1 asequenceinE such that there are B (E),U 0(E) such that for all >0 there is n() ∈ N with (T − Tn)(U) ⊆ B for all n ≥ n(). Then there is n0 ∈ N such that Tn is an isomorphism onto E for all n ≥ n0. Before we begin the proof we would like to note the following remark. ∞ Remark 2.12. If (Tn)n=1 and T satisfy the assumption of the proposition, then Tn converges to T in the sense of Mackey in Lb(E); in particular, Tn converges to T in Lb(E)asn tends to ∞. Proof. Since T is an isomorphism, V := T (U) is a neighbourhood in E and for every >0thereexistsn() ∈ N such that for every n ≥ n()wehave −1 −1 (I − TnT )(V ) ⊆ B.Thisimpliesthat,forn ≥ n(1), the operator (I − TnT )is ⊆ 1 ≥ bounded. There is 0 > 0 such that 0B 2 V .Putn0 := n(0). If n n0,we − −1 ⊆ 1 −1 have (I TnT )(V ) 2 V . We can apply Theorem 2.10 to conclude that TnT is an isomorphism onto E.ThisimpliesthatTn is an isomorphism for n ≥ n0. References

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Departamento de Matematica´ Aplicada, ETS Arquitectura, Universidad Politecnica´ de Valencia, E-46071 Valencia, Spain E-mail address: [email protected] Departamento de Matematica´ Aplicada, Fac. Informatica, Universidad Politecnica´ de Valencia, E-46071 Valencia, Spain E-mail address: [email protected]

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